Metric properties of incomparability graphs with an emphasis on paths

We describe some metric properties of incomparability graphs. We consider the problem of the existence of infinite paths, either induced or isometric, in the incomparability graph of a poset. Among other things, we show that if the incomparability graph of a poset is connected and has infinite diameter then it contains an infinite induced path and furthermore if the diameter of set of vertices of degree at least 3 is unbounded the graph contains as an induced subgraph either a comb or a kite. This result allows to draw a line between ages of permutation graphs which are well quasi order and those which are not.

[1]  R. A. Dean,et al.  Natural Partial Orders , 1968, Canadian Journal of Mathematics.

[2]  Mark E. Watkins,et al.  Infinite paths that contain only shortest paths , 1986, J. Comb. Theory, Ser. B.

[3]  Imed Zaguia,et al.  On Minimal Prime Graphs and Posets , 2008, Order.

[4]  Djamila Oudrar.  Sur l'énumération de structures discrètes, une approche par la théorie des relations , 2016 .

[5]  David Kelly.  The 3-Irreducible Partially Ordered Sets , 1977, Canadian Journal of Mathematics.

[6]  T. Gallai.  Transitiv orientierbare Graphen , 1967 .

[7]  E. Szpilrajn.  Sur l'extension de l'ordre partiel , 1930 .

[8]  Rudolf Halin,et al.  The structure of rayless graphs , 1998 .

[9]  R. P. Dilworth,et al.  A DECOMPOSITION THEOREM FOR PARTIALLY ORDERED SETS , 1950 .

[10]  R. Fraïssé.  L'Intervalle En Theorie Des Relations; Ses Generalisations Filtre Intervallaire Et Cloture D'Une Relation , 1984 .

[11]  William T. Trotter,et al.  Characterization problems for graphs, partially ordered sets, lattices, and families of sets , 1976, Discret. Math..

[12]  M. Pouzet,et al.  Graphs containing finite induced paths of unbounded length , 2020, ALGOS.

[13]  P. Ille.  L'Intervalle en théorie des relations , 1989 .

[14]  Vadim V. Lozin,et al.  Bipartite induced subgraphs and well‐quasi‐ordering , 2010, J. Graph Theory.

[15]  P. Fishburn.  Intransitive indifference with unequal indifference intervals , 1970 .

[16]  William T. Trotter,et al.  Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures , 1993, Discret. Math..

[17]  Vadim V. Lozin,et al.  Canonical Antichains of Unit Interval and Bipartite Permutation Graphs , 2011, Order.

[18]  G. A. Hedlund,et al.  Symbolic Dynamics II. Sturmian Trajectories , 1940 .

[19]  Imed Zaguia,et al.  Critically prime interval orders , 2008, Discret. Math..

[20]  Maurice Pouzet,et al.  Ensemble ordonné universel recouvert par deux chaines , 1978, J. Comb. Theory, Ser. B.

[21]  Norbert Polat.  Graphs without isometric rays and invariant subgraph properties, I , 1998, J. Graph Theory.

[22]  Vadim V. Lozin,et al.  Boundary Properties of Well-Quasi-Ordered Sets of Graphs , 2013, Order.

[23]  Wojciech A. Trybulec.  Partially Ordered Sets , 1990 .